All kinds of video file formats are used on the web including Quicktime. This problem is not really a video question as much as it is a web programming question. "What is the best way to embed video on my webpage".
The real problem you face is the size of the videos and their ultimate download speed for the people you hope will enjoy them. There are tools that are used for this. Some that come to mind are the built-in Premiere export utilities, Media Cleaner, and the just-released Sorenson Squeeze. (If anyone recommends others, please jump in, I'm not the expert on this!). Converting between quicktime, avi, mpeg, and others may turn out to be important but it will not fix your immediate problem.
What you need to do is (1) pick a video format that supports the way you want the video to be seen on the web, (2) find a way to convert the video you have into that format at the right size, and (3) make sure you know how to transfer it to the hosting web server properly. I have some experience with this, so you could look at my site and see if it gives you any ideas.
If you continue to struggle with this, perhaps you can provide more specific details so I can better help.